Question

Natasha is \[x\] years old and her mother is \[{{x}^{2}}\] years old. When her mother becomes \[11x\] years old then Natasha becomes \[{{x}^{2}}\] years old. Find their present ages.

Answer 1

Hint: We solve this problem by assuming that it took some \[y\] years so that Natasha's mother became \[11x\] years old. Then we use the given conditions that after \[y\] years Natasha mother age became \[11x\] and the age of Natasha became \[{{x}^{2}}\] where we get two equations of two variables. Then we can solve it by substituting the value of one variable from one equation in another.

Complete step by step answer:
We are given that the present age of Natasha as \[x\] and the present age of her mother as \[{{x}^{2}}\]
We are also given that when her mother becomes \[11x\] years old then Natasha becomes \[{{x}^{2}}\] years old.
Let us assume that it took \[y\] years so that Natasha’s mother's age became \[11x\] years.
Here, we can modify the statement as after \[y\] years Natasha’s mother age became \[11x\] from \[{{x}^{2}}\]
Now, by converting the above statement into mathematical equation we get
\[\begin{align}
  & \Rightarrow {{x}^{2}}+y=11x \\
 & \Rightarrow y=11x-{{x}^{2}}.....equation(i) \\
\end{align}\]
Now, let us take the Natasha age.
We are given that the condition that after \[y\] years Natasha age became \[{{x}^{2}}\] from \[x\]
By converting the above statement into mathematical equation we get
\[\Rightarrow x+y={{x}^{2}}\]
Now, by substituting the value of \[y\] from equation (i) in above equation we get
\[\begin{align}
  & \Rightarrow x+11x-{{x}^{2}}={{x}^{2}} \\
 & \Rightarrow 2{{x}^{2}}-12x=0 \\
 & \Rightarrow 2x\left( x-6 \right)=0 \\
\end{align}\]
We know that if \[a\times b=0\] then either of \[a,b\] will be zero.
By using the above condition let us take the first term then we get
\[\begin{align}
  & \Rightarrow 2x=0 \\
 & \Rightarrow x=0 \\
\end{align}\]
We know that the age of a person can never be zero.
Now, let us take the second term then we get
\[\begin{align}
  & \Rightarrow x-6=0 \\
 & \Rightarrow x=6 \\
\end{align}\]

Therefore we can say that the present age of Natasha is 6 years and the present age of her mother is 36 years.

Note: We are given that when her mother becomes \[11x\] years old then Natasha becomes \[{{x}^{2}}\] years old
Here, the above statement is considered after some years of present age. Then we get the equations as
\[\Rightarrow y=11x-{{x}^{2}}.....equation(i)\]
\[\Rightarrow x+y={{x}^{2}}\]
But students may do mistake that they consider the statement as present and take the equation as
\[\Rightarrow 11x={{x}^{2}}\]
But, this gives the wrong answer because the ages \[11x\] and \[{{x}^{2}}\] are taken with respect to present ages.